Coupling effects of surface energy, strain gradient, and inertia gradient on the vibration behavior of small-scale beams

Abstract

This paper develops three surface energy-enriched gradient elastic beam models, respectively, in the contexts of Euler-Bernoulli, Timoshenko, and Levinson-Brickford-Reddy kinematic hypotheses to investigate the coupling effects of surface energy, strain gradient, and inertia gradient on the vibration behavior of small-scale beams. Modified strain-inertia gradient elasticity theory and Gurtin-Murdoch surface elasticity theory are combined to capture three types of small-scale effects. The equations of motion and consistent boundary conditions are derived by using the Euler-Lagrange variational principle. To analyze the general free vibration problems of three non-classical beam models, we construct the related differential quadrature finite elements according to two kinds of differential quadrature-based geometric mapping schemes. The efficacy of our theoretical models and numerical solution methods is established by comparing the degenerated results with the reported ones. Finally, we apply the newly developed models to investigate the size-dependent behavior of freely vibrating small-scale beams. It is revealed that the coupling effects of three physical factors can result in not only the stiffening or softening characteristic of vibration frequencies but also the significant change in the vibration mode shapes. Besides, the introduction of the strain gradient effect may induce a boundary layer for small-scale beams with at least one end clamped.